Coleman Index

In 1971, three indices of a priori voting power were introduced by James S. Coleman: the Prevent Action index (), the Initiate Action index (*), and the Index of Collectivity to Act (A). The first two indices, suitably normalized, coincide with both the normalized Banzhaf index and the normalized Penrose index (see Banzhaf Voting Power Measure, Penrose Voting Power Measure). However, the Coleman indices have a greater conceptual value because they measure different aspects of the power to make decisions and to veto.

The definitions of the three Coleman indices (in absolute and normalized versions) are presented here and illustrate the links between the other previously mentioned indices. An example follows.

The Index of Collectivity to Act (A) measures the ability of a given collectivity to make decisions, that is, the potentiality that an act or bill has of being passed. In the collectivity of n members, there are 2n possible coalitions. The Index of Collectivity to Act is the quotient between the number of winning coalitions ω (i.e., the number of voting outcomes that lead to a decision) and 2”:

[Page 118]The essential element in the construction of the other two indices is the concept of “swing.” A swing for the i-th member of the collectivity is a pair of coalitions (S, S\ i}) such that S is a winning coalition and S\ i is losing. The number of swings of each member i is denoted by ci.

The Prevent Action index (γ) is a measure of the blocking power of each member in the given collectivity. For each member i, the Prevent Action Index γi is the quotient between the number of the member's swings ci and the number of the winning coalitions of the collectivity:

The Initiate Action index (*) is a measure of the power of each member to have his proposals accepted by the collectivity. For each member I, the Initiate Action index γi* is the quotient between the number of the member's swings ci and the number λ = 2n− ω of the losing coalitions of the collectivity, that is, of the outcomes that do not produce a decision:

The absolute Banzhaf index β′ coincides with the Penrose index r. Both the prevent action and the initiate action indices can be regarded as rescalings of β′ (and then of r). In fact,

In particular, both γ and γ* coincide with β′(and then with r) if the number of winning coalitions is equal to the number of the loosing coalitions (i.e., ω = λ = 2n−1) because in this case there is no difference between the power to prevent action and the power to initiate action.

Moreover, the absolute Banzhaf index β′ (and then r) coincides with the harmonic mean of and *:

Dan S. Felsenthal and Moshé Machover classified A,, and * as well as the (not normalized) indices of Banzhaf and Penrose, in the group of the I-power indices (power as influence).

For both and *, the sum extended to all the n members in general is greater than one. Then the normalized indices are

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